Proper and improper density matrices and the consistency of the ...

Proper and improper density matrices and the consistency of the Deutsch model for Local Closed Timelike Curves Augusto C´esar Lobo Physics Department, Federal University of Ouro Preto

arXiv:1110.6361v3 [quant-ph] 13 Mar 2012

Ismael Lucas de Paiva and Pedro Ruas Dieguez Physics Department, Federal University of Minas Gerais

Abstract We discuss the concept of proper and improper density matrixes and we argue that this issue is of fundamental importance for the understanding of the quantum-mechanical CTC model proposed by Deutsch. We arrive at the conclusion that under a realistic interpretation, the distinction between proper and improper density operators is not a relativistically covariant notion and this fact leads to the conclusion that the D-CTC model is physically inconsistent.

I.

adding some extent of non-linearity to its ax-

INTRODUCTION

ioms and surprisingly it turned out to be a Though quantum mechanics has been in- very difficult enterprise. Almost immediately disputably recognized for a long time now as it was pointed out that this would lead to sumuch as a counter-intuitive theory as it is an perluminal communication or signaling [4, 5]. extraordinarily effective one - it may be said As is well known, this leads to paradoxes and that the issue of its “rigidity” as a physical is considered by most physicists to be nonmodel has arisen only in more recent times physical. [1]. The “rigidity” of a theory can be thought as a qualitative measure of how much the en-

In fact, in 1949, G¨odel found a solution

tire theoretical construction is somehow con- of general relativistic field equations that strained by its principles in the sense that exhibits closed time-like curves (CTCs) [6]. by tampering with one of them would “bring Most physicists at the time (including Eindown” the whole structure, making the the- stein) found this result quite disturbing, but ory incoherent or inconsistent.

disregarded it as probably nonphysical, pre-

In 1989, Weinberg [2, 3] initiated an in- ferring to believe maybe that a future more vestigation on how far one could modify complete understanding of the physics inordinary quantum mechanical principles by volved should somehow rule out such kind of 1

a solution.

obey: ρˆCT C = trCR [Uˆ (ˆ ρ(i) ⊗ ρˆCT C )Uˆ † ]

Yet, in 1991, Deutsch presented a compu-

(1)

The physical meaning of the above equa-

tational analysis of the problem – both clas- tion is clear. The CR system (Alice) comes sical and quantum mechanical [7]. He ad- close enough to the CTC system (Bob), interdressed the inherent free-will kind of prob- acts with it in a limited region of space-time lems that stems from grandfather-like para- and becomes part of a larger entangled state ρ(i) ⊗ ρˆCT C )Uˆ † under the interaction moddoxes in a more technical and less anthropo- Uˆ (ˆ morphic way than usual. He assumed the ex- eled by the global unitary operator Uˆ . After a istence of a region of space-time (CTC) which while, the system moves away and Bob’s sysviolates the usual chronological respecting tem is obtained by partial tracing out Alice’s space-time (CR) and considered a compu- system. The above equation is a way to imtational circuit where one or more bits or pose that the output state is the same as the qubits may enter the CTC and travel back input one, obeying the CTC criterion with in time meeting itself at an earlier moment. no paradoxes as Deutsch showed that there He also conceived that a CR system may in- is always at least one solution (there may be teract with the CTC system by some unspec- more than one.) Note that ρˆCT C depends on (i) ified unitary interaction. He showed that, in the initial state ρˆ and on the unitary opergeneral, classical computational logic is in- ator Uˆ (the interaction). Alice’s system will

consistent with the existence of CTCs in the have changed to sense that not all possible input states have a consistent solution – this is the technical way of stating the free-will problem. Yet, he sug-

ρˆ(f ) = trCT C [Uˆ (ˆ ρ(i) ⊗ ρˆCT C )Uˆ † ]

(2)

The two equations above clearly imply a

gested a quantum mechanical model for sys- non-linear evolution for Alice’s system betems (allowing for mixture states), that ex- cause (2) means that ρˆ(f ) depends both on hibits output solutions for every input state, ρˆ(i) and ρˆCT C , (and on the interaction) but seemingly circumventing the paradoxes. Let the latter also depends on ρˆ(i) . This feature ρˆ(i) be the initial state of the chronology re- is a novel ingredient that goes beyond the specting system and ρˆCT C the state captured linear evolution described by Schr¨odinger’s in the CTC loop, then he proposed the fol- equation. Since Deutsch’s proposal, many lowing self consistent relation that ρˆCT C must results appeared in the literature where it 2

was argued that quantum mechanics together improper density matrices in quantum mewith Deutsch’s model for closed time-like chanics that are crucial to our main conclucurves (D-CTC) is more powerful than or- sions. In Section IV, we briefly review the dinary quantum physics. Claims as cloning results in [10] that allows the discrimination of quantum states [8], solving NP-complete of non-orthogonal states with the Deutsch problems in polynomial time [9] and distin- model and how this implies that the D-CTC guishing non-orthogonal states [10] have been model is not consistent. In Section V, we reported.

conclude our work with a discussion about

In this paper we discuss the fact that the physical meaning of these results and we under a certain interpretation of quantum also set stage for further work. mechanics, the ability to distinguish nonorthogonal states leads directly to inconsistencies in the Deutsch protocol for quantum systems traversing CTCs.

We argue

that the concept of a density matrix to be proper or improper is a relativistically noncovariant notion and that this result leads us to the above conclusion after we examine in detail a specific instance of a protocol for nonorthogonal state discrimination presented by Brun et al.

II.

NON-DISCRIMINATION OF NON-

ORTHOGONAL

STATES

IN

ORDI-

NARY QM

The celebrated 1935 EPR paper showed, for the first time, what Einstein coined as “spooky action at distance” of Bell-like states even though it was early recognized that it was impossible to use entangled states for superluminal communication [11]. Suppose two

The paper is structured as follows: In Section II, we will briefly review the nonlocal properties of a EPR-like pair of qubits and discuss why by the ordinary inter-

parties, Alice and Bob meet to establish an interaction between their qubits to create a maximal entangled pair of qubits in the anticorrelated form

pretation of quantum mechanics, the nondistinguishability of non-orthonormal states protects the theory from superluminal com-

1 |Ψi = √ (|z+i ⊗ |z−i − |z−i ⊗ |z+i) (3) 2

munication despite the non-local information After this, they go apart and each one has contained in the entangled states. In Sec- access only to his own qubit. Both parts will tion III, we discuss some interpretational is- describe their own system by the density masues on the difference between proper and trix 3

0 to the z direction and 1 to the x direcρˆAlice = ρˆBob

1 = Iˆ 2

(4) tion. By choosing which direction to measure her qubit, she would collapse the whole

This means that (3) can be written as

system instantaneously (in some inertial ref1 |Ψi = √ (|ˆ ni ⊗ |−ˆ ni − |−ˆ ni ⊗ |ˆ ni) 2

(5) erence system). Suppose she chooses to send the 0 bit and

Where the state |ˆ n (θ, ϕ)i defined as

after her z measurement, she “collapses” her

|ˆ n (θ, ϕ)i = cos (θ/2) |z+i + eiϕ sin (θ/2) |z−i state to |z+i. There is no way that Bob can (6) receive this classical piece of information. If is projected to an arbitrary point with (θ, ϕ) he chooses to measure the z direction, he will coordinates on the surface of the Bloch certainly obtain |z−i, but there is no way for sphere. Note that Hilbert space orthonormal- him to know if his collapsed pure state was in ity of two states means that the vectors are the x or z directions without receiving inforprojected to antipodal points on the Bloch mation from Alice in the usual (subluminal) sphere. To convey superluminal information, manner. The intrinsic randomness of quanone part should be able to distinguish differ- tum mechanics avoids superluminal commuent directions in the Bloch sphere geometry, nication. which means the ability to discriminate nonorthogonal states. Thus, Alice may “collapse” the mixed state by measuring her state in any direction of the Bloch Sphere, transforming non-locally Bob’s description to a pure state. But since we may assume that Bob is far away, he has no way to find out about Alice’s measurement without receiving a message from her. Alice and Bob could agree before hand to measure only over two different directions, let us say

By the other way around, the ability of

the x and z directions. Alice could hope- discrimination of non-orthogonal states leads lessly try to code one classical bit of infor- necessarily to signaling as can be easily seen. mation in these distinguished directions, by Most papers on no-signaling and quantum assigning logical values to them, for instance, mechanics discuss this issue in terms of the 4

no-cloning theorem ([12–14]) because both four state vectors of their common alphabet, these abilities are actually equivalent [15]. Alice will clearly be able to communicate a Curiously, we were incapable of finding in classical bit in a superluminal way by choosthe literature any explicit discussion of the ing in which of the two agreed directions she no-signaling property of quantum mechanics performs her measurement. in terms of the indistinguishability of nonorthogonal states. This issue seems to have solely been discussed in terms of the nocloning theorem. Maybe this is because signaling is such an obvious consequence of nonorthogonal state distinguishability that most

III.

SOME INTERPRETATIONAL IS-

SUES OF QUANTUM MECHANICS

A.

The interpretation of the state

vector

authors just take it for granted. On possible exception is Gisin’s 1989 paper where

The picture of entanglement presented in

he refutes Weinberg’s attempt to construct the last Section goes along with the most a non-linear quantum mechanical theory [4]. usual interpretation of quantum physics. Indeed in his paper, he manages to ultimately This interpretation (let us agree to call it distinguish non-orthogonal states, but only a realistic interpretation) views the entanafter a quite elaborate construction where glement phenomenon as exhibiting genuine there must be a third party besides Alice non-local properties even if this non-locality and Bob that must continuously provide a does not lead to signaling. In this interprestream of entangled qubits, each one sent to tation, the entangled state (5) can be seen one of the other parts. (Alice and Bob). We as a non-causal channel connecting two disshow in our above analysis that any “non- tant observers such that if a pair of measureunitary machine” capable of discriminating ments are performed by Alice and Bob (one non-orthogonal states immediately implies in measurement each - at space-like separated superluminality, a result that seems quite ob- events) then it follows that there is no convious, but which seems that has not been ex- sistent way to assign any causal relation beplicitly stated anywhere before in the litera- tween them. ture. One may say that in this view, one thinks In fact, going back to the EPR protocol of the state vector (or its projection onto the discussed above, if Bob has access to some space of rays) as a real objective property device capable of discriminating between the of the system. These quantum channels are 5

behind many of the modern applications of hidden variable correlations and showed that quantum information as quantum teleporta- the dispute was physically verifiable. One tion and cryptography.

may even make the case that this was the

A very different view is taken by some moment when the embryo of modern quanphysicists like [16] or [17] for example. This tum information science was laid. Our opinapproach to quantum mechanics may be de- ion is that indeed for linear quantum mechanscribed as epistemic in the sense that one as- ics, there may not be any physical differences signs to the state vector the subjective prop- between these two approaches. But if any erty of describing only the knowledge of an non-linearity is introduced as some physicists observer about the system. For those who expect for a full consistent quantum theory of subscribe to this interpretation, there is no gravity (see [18] and [19] for opposite argusuch thing as non-local phenomena in quan- mentations on this issue), then the situation tum mechanics, because the non-local col- will probably be very different as we argue in lapse of the global state vector is a non- the next sections. physical process. In [17], for instance, the authors conclude that the Deutsch scheme manages only to conceal the paradoxes in-

B.

The interpretation of the density

matrix

volved in time-travelling instead of resolving them.

There are two very different approaches to

Yet, regardless of these different inter- the density matrix concept. The first may be pretations, for linear quantum mechanics, it called the text-book concept of a density mais noncontroversial that one arrives at the trix and it is a reminiscent of the historical same physical predictions. One could wonder way that statistical concepts were introduced if these subtle philosophical distinctions are in classical physics. One is given a large enthen actually physically relevant. But some- semble of N identical quantum systems and thing quantum mechanics has taught us dur- one supposes that the ensemble can be paring the last century is that one should be spe- titioned into many sub-ensembles labeled by cially careful before jumping to such a conclu- an index α = 1, 2, ...m where m is the number sion. After all, the Bohr-Einstein debate was of different sub-ensembles α characterized by generally considered (for three decades) as the fact that every quantum system that bebeing of a philosophical character before Bell longs to it is in a pure state |ψα i. Now supintroduced in 1964 his inequalities for local pose further that an observer may “pick” one 6

ˆ be again an arbitrary observable of Alsystem from the ensemble in a random way. Let O Notice that this randomness is purely “classi- ice’s subsystem, then she can define a density cal” in some sense. The probability that the matrix ρˆ|Ψi by the following equation observer picks up a system from a subensemble α is clearly Pα = nα /N where nα is the number of systems in sub-ensemble α. Of course it clearly holds that the probability

  ˆ ⊗ Iˆ |Ψi = tr ρˆ|Ψi O ˆ hΨ| O

ˆ (for all O)

The above equation defines a mapping from the space of rays (the projective space

Pα is automatically normalized as it indeed of the full quantum space W ) to the space ˆ its must be. Given an arbitrary observable O, of linear operators in W (a) . This (non-linear) ˆ expectation value for |ψα i is hψα | O |ψα i and mapping is called the partial trace and it can if the observer repeats the procedure many be conducted in the same manner for Bob’s times, the average expectation value is clearly subspace. This definition of density matrix is   X ˆ |ψα i = tr ρˆO ˆ Pα hψα | O known as an improper mixture and it is comα

monly thought as an ontological description

where ρˆ =

X

of a mixed state in some interpretations and

Pα |ψα i hψα |

should then be seen as essentially different

α

is defined as the density matrix. This defini- from the former case. tion is easily seen as an epistemic definition What is remarkable is that mathematiwhere one considers Pα a classical probabil- cally, both definitions lead to equivalent deity distribution that is subjective in the sense scriptions. Both (proper and improper mixthat it reveals the “classical ignorance” of the tures) are hermitian, positive and unit trace observer about which pure state |ψα i the sys- operators. Some authors deny the existence tem actually belongs to. This epistemic defi- of proper mixtures in the sense that all dennition is known also as a proper density ma- sity matrices can be thought as resulting from trix [20].

a partial trace. There is some controversy on

A second conception of density matrix has this matter. (See [21] and [22] for opposite a purely quantum mechanical origin. Given opinions on this issue.) But consider now the an entangled state |Ψi ∈ W = W (a) ⊗ W (b) following two experiments: of two subsystems (Alice and Bob’s subsystems), suppose (after the global entangled

1. Alice tosses a fair coin in her lab to de-

state has been created locally) each one has

cide if she produces a pure |z+i or |z−i

physical access only to its own subsystem.

state and then sends the state to Bob.

7

2. Alice initially produces an entangled 2- where {ˆ πj } is a family of one-dimensional qubit state as in (5) and then measures projection operators over an orthonormal baˆ j } is a family of unitary one of the qubits in the z direction and sis {|uj i} of W and {O ˆ k |ψk i = |uk i. Brun operators such that O

sends the other qubit to Bob. It is clear that both preparations must lead to physically indistinguishable states for Bob (at least for usual linear quantum mechanics) represented both by the same proper density matrix given by (4). If one introduces elements of non-linearity as is the case of the D-CTC prescription, things are not as simple

et al have shown that it is always possible ˆ j } such that there is only to find a set {O one single self-consistent solution given by ρˆCT C = π ˆs . This allows the existence of a bijective (non-linear) map between a set of non-orthogonal states in W and a set of orthogonal states {|uj i} which implies the discrimination of the elements of C.

as we shall see in the next Section.

With this result, one may approach again IV.

NON-ORTHOGONAL

STATE

DISCRIMINATION PROTOCOL WITH

the EPR problem and conclude that superluminal communication seems to be possible. In fact, suppose Alice and Bob share

DEUTSCH CTC’S

a single pair of qubits in the Bell state given In [10], Brun et al exhibited a quan- by (5) where the first qubit stays in Alice’s tum computational protocol with access to possession and the second qubit travels with a Deutsch CTC that allows discrimination of Bob. Suppose they have agreed previously over the code described in Section I. Bob now

arbitrary non-orthogonal state vectors.

Let C = {|ψk i}, (k = 1, ...n) be a set must join his qubit with another qubit sysof n non-orthogonal normalized state vectors tem in a known state |z+i for instance. In belonging to a finite n-dimensional Hilbert this way, after Alice measures her qubit in space W of the CR system. Suppose that one of the two possible directions, Bob’s systhe input state is given by the following pure tem will be in a state |ξj i ⊗ |z+i, with the density matrix ρˆCR = π ˆ|ψs i = |ψs i hψs | cho- state |ξj i among the following four possibilsen from C. The protocol starts by applying ities: |ξ0 i = |z+i, |ξ1 i = |z−i, |ξ2 i = |x+i the swap operator on the W ⊗ WCT C system or |ξ3 i = |x−i. Bob then uses the D-CTC to distinguish these non-orthogonal states. The followed by the controlled Uˆ operator X input state is the pure state |ξj i ⊗ |z+i and ˆj Uˆ = π ˆj ⊗ O (7) the circuit will swap the CR and CTC system j 8

followed by the controlled unitary operation exhibit non-locality in the EPR experiments (7) with |u0 i = |z+i ⊗ |z+i, |u1 i = |z−i ⊗ [24]. |z+i, |u2 i = |z+i ⊗ |z−i, |u3 i = |z−i ⊗ |z−i Notice that Alice and Bob can arrange ˆ i (|ξi i ⊗ |z+i) = |ui i. In this way, Bob things such that for a certain moment just and O manages to know which direction Alice de- before Bob interacts with the D-CTC, Alice cided to measure and her classical bit is com- measures the system in the way we described municated with infinite speed.

so that Bob is sure that his state is pure. His

In a recent paper [23], the authors reach a description would be given by a proper dendifferent conclusion. They show (as Deutsch sity matrix so that he can treat his state in assumes in his seminal paper) that quantum the same way that Brun et al and Ralf et correlations are lost in the D-CTC circuit by al treat the states from the non-orthogonal using a “infinite loop” equivalent circuit for- alphabet of states chosen between him and mulation where the problem is seen by the Alice. At this point - a physicist that beperspective of the CTC system that is “cap- lieves in a radically epistemic interpretation tured for eternity” in the loop. We subscribe of quantum mechanics might reach a differto this view but the authors then make a ent conclusion. He may say that there is no claim that we believe is mistaken: they claim difference between proper and improper mixthat this fact implies that superluminal phe- tures and that they should be treated in the nomena cannot happen. It seems that they same way. This is the “linearity trap argubelieve that the “collapse” of the global state ment” put forward in [25] but convincingly of a entangled state must happen only locally. refuted in [26] with a sound argument of veriBut this belief is extraneous to the usual view fiability for any non-linear evolution. Indeed, of quantum mechanics. And neither have we Ralph et al also subscribe to Cavalcanti and found elements in Deutsch’s original paper Menicucci’s opinion but seem not to recogthat supports such extreme notion. To sub- nize that the same argument leads to signalscribe to this idea would be the same as in- ing by the argument that we present here. troducing new elements in quantum physics

An important point that must be made is

that indicate some kind of physical medium that of taking seriously the fact that the meathat would propagate the “wave-function col- surements are space-like separated events. lapse” with finite speed.

This is contrary This means that there is a reference system

to everything we know about quantum me- for which Alice has not yet “collapsed” the chanics including the well known results that global state and in this case indeed Bob will 9

insert his qubit as an improper state (4).

some fundamental questions about the model

This result implies that the concept of that may be addressed.

For instance, in

proper and improper mixtures is not a rela- Deutsch’s original scheme, the unitary opertivistically covariant notion. But one should ator to be applied in the interaction with the note that for usual linear quantum mechanics CTC is supposed to be arbitrary. Is this a this is irrelevant because both concepts are reasonable supposition on physical grounds? equivalent. In this case we have a completely After all, shouldn’t one expect that under the different physical picture. We agree that in- very likely extreme physical conditions that deed in this scenario, the D-CTC should de- material particles suffer under closed timestroy all quantum correlations and the output like curves as their world-lines, that the physwould again be the maximum entangled state ically possible unitary quantum evolutions should be constrained by the rules of a (still

(4).

How can this be possible? The only con- unknown) consistent quantum theory of gravceivable answer to this question is that it itation? cannot be. We summarize our analysis in

A full description of the largest class of

the following way: In one reference system, unitary evolutions that not allow arbitrary Bob’s state is pure and he manages to re- non-orthogonal state discrimination might ceive superluminal communication which ul- turn out to be rewarding in the sense that timately leads to those same paradoxes that this information could be used as a clue for Deutsch was originally trying to prevent and what kind of interaction are or not permitso it is then inconsistent. In another refer- ted in such extreme scenarios. Even if such ence state, his state is an improper mixture a programme could be carried out successand the signaling protocol fails. But this is fully, the issue of the relativistically noncoinconsistent with the first scenario. We con- variance of the density matrix concept and clude then that Deutsch’s CTC model itself the resulting lack of inner consistency that it is overall inconsistent.

implies would also have to be tackled – see [27] for a recent discussion on related mat-

V.

ters. Maybe there is a manner of restraining

CONCLUDING REMARKS

the set of unitary operations that could at Does this analysis imply necessarily the the same time guarantee an equivalent physconclusion that the Deutsch CTC model is ical description for all reference frames. Of irreparably inconsistent after all? There are course, the fact that the many tasks that have 10

been recently claimed to be possible of imple- fore [33], [34]. It would not be surprising mentation with Deutsch CTCs may be seen if Aharonov’s concepts of modular variables as a sign that this model is inherently incon- and weak values turn out to play an essential sistent after all.

role on this issue [35], [36]. It is our opinion

It should also be noted that there is an that the importance of research on this topic alternative model to Deutsch’s CTC. The is that of presenting toy models so that one so called post-selection CTC (see [28], [29], may probe theoretically a very elusive long[30], [31] and [32]) is a recent example of a sought consistent quantum theory of gravmodel that allows non-orthogonal discrimi- ity. Our work also suggests that the connation only for a set of linearly independent struction of such a future theory may require states. Notice that this clearly forbids the that some important foundational issues (as implementation of our signaling protocol be- the intrinsic difference between proper and tween Alice and Bob because the set C of improper density matrices) be addressed and alphabet states is linearly dependent. This adequately resolved first. seems to imply that the P-CTC model may indeed be more appropriate physically than

Acknowledgments

the D-CTC model. It is our intention to publish in the near future a paper specifically on the P-CTC model in quantum mechanics.

A.C.Lobo acknowledges financial support from NUPEC-Gorceix Foundation and

That the ability to post-select may present I.L.Paiva acknowledges financial support counter-intuitive non-local time properties from FAPEMIG. All authors acknowledge fifor quantum mechanics has been noticed be- nancial support from CNPq.

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